C**********************************************************************
C
C Copyright (C) 1992 Roland W. Freund and Noel M. Nachtigal
C All rights reserved.
C
C This code is part of a copyrighted package. For details, see the
C file "cpyrit.doc" in the top-level directory.
C
C *****************************************************************
C ANY USE OF THIS CODE CONSTITUTES ACCEPTANCE OF THE TERMS OF THE
C COPYRIGHT NOTICE
C *****************************************************************
C
C**********************************************************************
C
C This file contains the routine for the QMR algorithm for
C symmetric matrices, using the three-term recurrence variant of
C the Lanczos algorithm without look-ahead.
C
C**********************************************************************
C
SUBROUTINE SSQMX (NDIM,NLEN,NLIM,VECS,TOL,INFO)
C
C Purpose:
C This subroutine uses the QMR algorithm to solve linear systems.
C It runs the algorithm to convergence or until a user-specified
C limit on the number of iterations is reached. It is set up to
C solve symmetric systems starting with identical starting vectors.
C
C The code is set up to solve the system A x = b with initial
C guess x_0 = 0. Here A x = b denotes the preconditioned system,
C and it is connected with the original system as follows. Let
C B y = c be the original unpreconditioned system to be solved, and
C let y_0 be an arbitrary initial guess for its solution. Then:
C A x = b, where A = M_1^{-1} B M_2^{-1},
C x = M_2 (y - y_0), b = M_1^{-1} (c - B y_0).
C Here M = M_1 M_2 is the preconditioner.
C
C To recover the final iterate y_n for the original system B y = c
C from the final iterate x_n for the preconditioned system A x = b,
C set
C y_n = y_0 + M_2^{-1} x_n.
C
C The implementation does not have look-ahead, so it is less robust
C than the full version.
C
C Parameters:
C For a description of the parameters, see the file "ssqmx.doc" in
C the current directory.
C
C External routines used:
C single precision slamch(ch)
C LAPACK routine, computes machine-related constants.
C single precision snrm2(n,x,incx)
C BLAS-1 routine, computes the 2-norm of x.
C subroutine saxpby(n,z,a,x,b,y)
C Library routine, computes z = a * x + b * y.
C single precision sdot(n,x,incx,y,incy)
C BLAS-1 routine, computes y^H * x.
C single precision zqmxom(n)
C User-supplied routine, specifies the QMR scaling factors.
C subroutine srandn(n,x,seed)
C Library routine, fills x with random numbers.
C subroutine srotg(a,b,cos,sin)
C BLAS-1 routine, computes the Givens rotation which rotates the
C vector [a; b] into [ sqrt(a**2 + b**2); 0 ].
C
C Noel M. Nachtigal
C May 25, 1993
C
C**********************************************************************
C
INTRINSIC ABS, FLOAT, AMAX1, SQRT, MAX0
INTRINSIC MOD
EXTERNAL SLAMCH, SNRM2, SAXPBY, SDOT, SRANDN, SROTG, SSQMXO
REAL SDOT
REAL SLAMCH, SNRM2, SSQMXO
C
INTEGER INFO(4), NDIM, NLEN, NLIM
REAL VECS(NDIM,7)
REAL TOL
C
C Miscellaneous parameters.
C
REAL DHUN, DONE, DTEN, SZERO
PARAMETER (DHUN = 1.0E2,DONE = 1.0E0,DTEN = 1.0E1,SZERO = 0.0E0)
C
C Local variables, permanent.
C
INTEGER IERR, ISN, ISNM1, ISNM2, IVN, IVNM1, IVNP1, IWN, IWNM1
SAVE IERR, ISN, ISNM1, ISNM2, IVN, IVNM1, IVNP1, IWN, IWNM1
INTEGER IWNP1, N, RETLBL, TF, TRES, VF
SAVE IWNP1, N, RETLBL, TF, TRES, VF
REAL DN, DNM1, DNP1, RHSN, SCSN, SCSNM1, SINN, SINNM1
SAVE DN, DNM1, DNP1, RHSN, SCSN, SCSNM1, SINN, SINNM1
REAL COSN, COSNM1, CSIN, CSINP1, MAXOMG, OMGN
SAVE COSN, COSNM1, CSIN, CSINP1, MAXOMG, OMGN
REAL OMGNP1, R0, RESN, RHON, RHONP1, SCVN, SCVNP1
SAVE OMGNP1, R0, RESN, RHON, RHONP1, SCVN, SCVNP1
REAL SCWN, SCWNP1, TMAX, TMIN, TNRM, UCHK, UNRM
SAVE SCWN, SCWNP1, TMAX, TMIN, TNRM, UCHK, UNRM
C
C Local variables, transient.
C
INTEGER INIT, REVCOM
REAL RHN, RHNM1, RHNM2, RHNP1, RHSNP1
REAL STMP, SCVNM1, SCWNM1
C
C Initialize some of the permanent variables.
C
DATA RETLBL /0/
C
C Check the reverse communication flag to see where to branch.
C REVCOM RETLBL Comment
C 0 0 first call, go to label 10
C 1 30 returning from AXB, go to label 30
C 1 60 returning from AXB, go to label 60
C 2 40 returning from ATXB, go to label 40
C
REVCOM = INFO(2)
INFO(2) = 0
IF (REVCOM.EQ.0) THEN
N = 0
IF (RETLBL.EQ.0) GO TO 10
ELSE IF (REVCOM.EQ.1) THEN
IF (RETLBL.EQ.30) THEN
GO TO 30
ELSE IF (RETLBL.EQ.60) THEN
GO TO 60
END IF
ELSE IF (REVCOM.EQ.2) THEN
IF (RETLBL.EQ.40) GO TO 40
END IF
IERR = 1
GO TO 80
C
C Check whether the inputs are valid.
C
10 IERR = 0
IF (NDIM.LT.1) IERR = 2
IF (NLEN.LT.1) IERR = 2
IF (NLIM.LT.1) IERR = 2
IF (NLEN.GT.NDIM) IERR = 2
IF (IERR.NE.0) GO TO 80
C
C Extract from INFO the output units TF and VF, the true residual
C flag TRES, and the left starting vector flag INIT.
C
VF = MAX0(INFO(1),0)
INIT = VF / 100000
VF = VF - INIT * 100000
TRES = VF / 10000
VF = VF - TRES * 10000
TF = VF / 100
VF = VF - TF * 100
C
C Extract and check the various tolerances.
C
TNRM = SLAMCH('E') * DTEN
TMIN = SQRT(SQRT(SLAMCH('S')))
TMAX = DONE / TMIN
IF (TOL.LE.SZERO) TOL = SQRT(SLAMCH('E'))
C
C Start the trace messages and convergence history.
C
IF (VF.NE.0) WRITE (VF,'(I8,2E11.4)') 0, DONE, DONE
IF (TF.NE.0) WRITE (TF,'(I8,2E11.4)') 0, DONE, DONE
C
C Initialize the wrapped indices.
C
ISNM1 = 5
ISN = ISNM1
IVN = 3
IVNP1 = IVN
IWN = 3
IWNP1 = IWN
C
C Set x_0 = 0 and compute the norm of the initial residual.
C
CALL SAXPBY (NLEN,VECS(1,IVN),DONE,VECS(1,2),SZERO,VECS(1,IVN))
CALL SAXPBY (NLEN,VECS(1,1),SZERO,VECS(1,1),SZERO,VECS(1,1))
R0 = SNRM2(NLEN,VECS(1,IVN),1)
IF ((TOL.GE.DONE).OR.(R0.EQ.SZERO)) GO TO 80
C
C Check whether the auxiliary vector must be supplied.
C
CSYM IF (INIT.EQ.0) CALL SRANDN (NLEN,VECS(1,IWN),1)
C
C Compute scale factors and check for invariant subspaces.
C
SCVNP1 = R0
CSYM SCWNP1 = SNRM2(NLEN,VECS(1,IWN),1)
SCWNP1 = SCVNP1
IF (SCVNP1.LT.TNRM) IERR = IERR + 16
IF (IERR.NE.0) GO TO 80
DNP1 = SDOT(NLEN,VECS(1,IWN),1,VECS(1,IVN),1) / ( SCVNP1 * SCWNP1
$)
IF ((SCVNP1.GE.TMAX).OR.(SCVNP1.LE.TMIN)) THEN
STMP = DONE / SCVNP1
CALL SAXPBY (NLEN,VECS(1,IVN),STMP,VECS(1,IVN),SZERO,VECS(1,IVN
$))
SCVNP1 = DONE
END IF
IF ((SCWNP1.GE.TMAX).OR.(SCWNP1.LE.TMIN)) THEN
STMP = DONE / SCWNP1
CSYM CALL SAXPBY (NLEN,VECS(1,IWN),STMP,VECS(1,IWN),SZERO,VECS(1,IWN))
SCWNP1 = DONE
END IF
RHONP1 = SCVNP1
CSINP1 = SCWNP1
SCVNP1 = DONE / SCVNP1
SCWNP1 = DONE / SCWNP1
C
C Initialize the variables.
C
N = 1
DN = DONE
COSN = DONE
RESN = DONE
COSNM1 = DONE
OMGN = SZERO
SCSN = SZERO
SCVN = SZERO
SCWN = SZERO
SCSNM1 = SZERO
SINN = SZERO
SINNM1 = SZERO
OMGNP1 = SSQMXO(N)
RHSN = OMGNP1 * R0
MAXOMG = DONE / OMGNP1
C
C This is one step of the classical Lanczos algorithm.
C
20 IVNM1 = IVN
IVN = IVNP1
IVNP1 = MOD(N,2) + 3
IWNM1 = IWN
IWN = IWNP1
IWNP1 = MOD(N,2) + 3
C
C Check whether D_n is nonsingular.
C
DNM1 = DN
DN = DNP1
IF (DN.EQ.SZERO) THEN
IERR = 8
GO TO 80
END IF
C
C Have the caller carry out AXB, then return here.
C CALL AXB (VECS(1,IVN),VECS(1,7))
C
INFO(2) = 1
INFO(3) = IVN
INFO(4) = 7
RETLBL = 30
RETURN
30 RETLBL = 0
C
C Compute H_{n-1,n} and build part of the vector v_{n+1}.
C
SCVNM1 = SCVN
CSIN = CSINP1
SCVN = SCVNP1
STMP = CSIN * DN / DNM1 * SCVNM1 / SCVN
CALL SAXPBY (NLEN,VECS(1,IVNP1),DONE,VECS(1,7),-STMP,VECS(1,IVNM1)
$)
C
C Have the caller carry out ATXB, then return here.
C CALL ATXB (VECS(1,IWN),VECS(1,7))
C
INFO(2) = 2
INFO(3) = IWN
INFO(4) = 7
RETLBL = 40
CSYM RETURN
40 RETLBL = 0
C
C Build part of the vector w_{n+1}.
C
SCWNM1 = SCWN
RHON = RHONP1
SCWN = SCWNP1
STMP = RHON * DN / DNM1 * SCWNM1 / SCWN
CSYM CALL SAXPBY (NLEN,VECS(1,IWNP1),DONE,VECS(1,7),-STMP,VECS(1,IWNM1))
C
C Compute H_{nn} and finish the new vectors.
C
RHN = SCVN * SCWN * SDOT(NLEN,VECS(1,IWN),1,VECS(1,IVNP1),1) / DN
CALL SAXPBY (NLEN,VECS(1,IVNP1),DONE,VECS(1,IVNP1),-RHN,VECS(1,IVN
$))
CSYM CALL SAXPBY (NLEN,VECS(1,IWNP1),DONE,VECS(1,IWNP1),-RHN,VECS(1,IWN))
C
C Compute scale factors and check for invariant subspaces.
C
IERR = 0
SCVNP1 = SNRM2(NLEN,VECS(1,IVNP1),1)
CSYM SCWNP1 = SNRM2(NLEN,VECS(1,IWNP1),1)
SCWNP1 = SCVNP1
RHONP1 = SCVN * SCVNP1
CSINP1 = SCWN * SCWNP1
RHNP1 = RHONP1
IF (SCVNP1.LT.TNRM) IERR = IERR + 16
IF (IERR.NE.0) GO TO 50
DNP1 = SDOT(NLEN,VECS(1,IWNP1),1,VECS(1,IVNP1),1) / ( SCVNP1 *
$SCWNP1 )
IF ((SCVNP1.GE.TMAX).OR.(SCVNP1.LE.TMIN)) THEN
STMP = DONE / SCVNP1
CALL SAXPBY (NLEN,VECS(1,IVNP1),STMP,VECS(1,IVNP1),SZERO,VECS(1
$,IVNP1))
SCVNP1 = DONE
END IF
IF ((SCWNP1.GE.TMAX).OR.(SCWNP1.LE.TMIN)) THEN
STMP = DONE / SCWNP1
CSYM CALL SAXPBY (NLEN,VECS(1,IWNP1),STMP,VECS(1,IWNP1),SZERO,VECS(1,IWNP1))
SCWNP1 = DONE
END IF
SCVNP1 = DONE / SCVNP1
SCWNP1 = DONE / SCWNP1
C
C The QMR code starts here.
C Multiply the new column by the previous omegas.
C Get the next scaling factor omega(i) and update MAXOMG.
C
50 RHNM1 = CSIN * DN * OMGN / DNM1
OMGN = OMGNP1
RHN = OMGN * RHN
OMGNP1 = SSQMXO(N+1)
RHNP1 = OMGNP1 * RHNP1
MAXOMG = AMAX1(MAXOMG,DONE/OMGN)
C
C Apply the previous rotations.
C
RHNM2 = SINNM1 * RHNM1
RHNM1 = COSNM1 * RHNM1
COSNM1 = COSN
SINNM1 = SINN
STMP = RHNM1
RHNM1 = COSNM1 * STMP + SINNM1 * RHN
RHN = -SINNM1 * STMP + COSNM1 * RHN
C
C Compute the rotation for the last element (this also applies it).
C
CALL SROTG (RHN,RHNP1,COSN,SINN)
C
C Apply the new rotation to the right-hand side vector.
C
RHSNP1 = -SINN * RHSN
RHSN = COSN * RHSN
C
C Compute the next search direction s_n.
C
ISNM2 = ISNM1
ISNM1 = ISN
ISN = MOD(N-1,2) + 5
STMP = SCSNM1 * RHNM2 / SCVN
CALL SAXPBY (NLEN,VECS(1,ISN),DONE,VECS(1,IVN),-STMP,VECS(1,ISNM2)
$)
SCSNM1 = SCSN
STMP = SCSNM1 * RHNM1 / SCVN
CALL SAXPBY (NLEN,VECS(1,ISN),DONE,VECS(1,ISN),-STMP,VECS(1,ISNM1)
$)
SCSN = SCVN / RHN
C
C Compute the new QMR iterate, then scale the search direction.
C
STMP = SCSN * RHSN
CALL SAXPBY (NLEN,VECS(1,1),DONE,VECS(1,1),STMP,VECS(1,ISN))
STMP = ABS(SCSN)
IF ((STMP.GE.TMAX).OR.(STMP.LE.TMIN)) THEN
CALL SAXPBY (NLEN,VECS(1,ISN),SCSN,VECS(1,ISN),SZERO,VECS(1,ISN
$))
SCSN = DONE
END IF
C
C Compute the residual norm upper bound.
C If the scaled upper bound is within one order of magnitude of the
C target convergence norm, compute the true residual norm.
C
RHSN = RHSNP1
UNRM = SQRT(FLOAT(N+1)) * MAXOMG * ABS(RHSNP1) / R0
UCHK = UNRM
IF ((TRES.EQ.0).AND.(UNRM/TOL.GT.DTEN).AND.(N.LT.NLIM)) GO TO 70
C
C Have the caller carry out AXB, then return here.
C CALL AXB (VECS(1,1),VECS(1,7))
C
INFO(2) = 1
INFO(3) = 1
INFO(4) = 7
RETLBL = 60
RETURN
60 RETLBL = 0
CALL SAXPBY (NLEN,VECS(1,7),DONE,VECS(1,2),-DONE,VECS(1,7))
RESN = SNRM2(NLEN,VECS(1,7),1) / R0
UCHK = RESN
C
C Output the trace messages and convergence history.
C
70 IF (VF.NE.0) WRITE (VF,'(I8,2E11.4)') N, UNRM, RESN
IF (TF.NE.0) WRITE (TF,'(I8,2E11.4)') N, UNRM, RESN
C
C Check for convergence or termination. Stop if:
C 1. algorithm converged;
C 2. there is an error condition;
C 3. the residual norm upper bound is smaller than the computed
C residual norm by a factor of at least 100;
C 4. algorithm exceeded the iterations limit.
C
IF (RESN.LE.TOL) THEN
IERR = 0
GO TO 80
ELSE IF (IERR.NE.0) THEN
GO TO 80
ELSE IF (UNRM.LT.UCHK/DHUN) THEN
IERR = 4
GO TO 80
ELSE IF (N.GE.NLIM) THEN
IERR = 4
GO TO 80
END IF
C
C Update the running counter.
C
N = N + 1
GO TO 20
C
C Done.
C
80 NLIM = N
RETLBL = 0
INFO(1) = IERR
C
RETURN
END
C
C**********************************************************************
C
REAL FUNCTION SSQMXO (I)
C
C Purpose:
C Returns the scaling parameter OMEGA(I).
C
C Parameters:
C I = the index of the parameter OMEGA (input).
C
C Noel M. Nachtigal
C March 30, 1993
C
C**********************************************************************
C
INTEGER I
C
SSQMXO = 1.0E0
C
RETURN
END
C
C**********************************************************************